Now focus on the second term in eq. (REF ). First, suppose that condition (1) of proposition REF  holds. Since \(\sup _T \operatorname{\mathbb {E}}{\varepsilon _{i, T} }^{\beta }<\infty \) , by lemma REF  we conclude that \(\theta _{N, N}-\vartheta _{N, N} = O_p({N^{1/\beta }}/{T^{p}})\) . By lemma REF , \({(\vartheta _{b-q, b, T}^{(s)} - \vartheta _{b, b, T}^{(s)}) }/{(\theta _F - U_F(b))} = O_p(1)\) . In addition, by corollary 1.2.10 in [1]} \({1}/{(\theta _F-U_F(t))}\)  is \(RV_{-\gamma }\) , so by lemma REF  we can write \({1}/{(\theta _F-U_F(t))} = t^{-\gamma } L(t)\)  for some slowly varying \(L\) . Since \(b=N^m\) , we obtain
\(\dfrac{ \theta _{N, N} - \vartheta _{N, N, T} }{ \vartheta _{b-q, b, T}^{(s)} - \vartheta _{b, b, T}^{(s)} } & = \left( \theta _{N, N} - \vartheta _{N, N, T}\right) \dfrac{ \theta _F - U_F(b)}{ \vartheta _{b-q, b, T}^{(s)} - \vartheta _{b, b, T}^{(s)}} \dfrac{1}{\theta _F- U_F(b)} \\& = O_p\left(\frac{N^{1/\beta }}{T^{p}}\right) O_p(1) O_p\left( \frac{1}{\theta _F-U_F(N^m)} \right) \\& = O_p\left(\frac{N^{1/\beta }}{T^{p}}\right) O_p\left( N^{-\gamma m} L(N^m) \right) . \) 
